Context

This article is about a technique which can come quite handy when one wants to build multiple time series models for the time series which have inherently heirarchical structure.

Time series can often be naturally disaggregated in a hierarchical structure using attributes such as geographical location, product type, etc. For example, the total number of Member of Parliaments(MPs) in a given election can come from different States and in turn given a particular State ,from different cities,districts and so forth.Such disaggregation imposes a hierarchical structure. We refer to these as hierarchical time series.

Another possibility is that series can be naturally grouped together based on attributes without necessarily imposing a hierarchical structure. For example the MPs in the above context can be filtered down also on the basic of Sex viz.Male ,Female and Others. Grouped time series can be thought of as hierarchical time series that do not impose a unique hierarchical structure in the sense that the order by which the series can be grouped is not unique.

I will try and focus more on the intuition rather than the mathematical details here but if one is interested in the same please refer here

Let’s plot some heirarchical time series below.This one pertains to Total quarterly visitor nights from 1998-2011 for eight regions of Australia:

Data Description

I will show below some graphs and tables pertaining to the Data at hand so as to expose it better.

Time Series Title Description
Sydney The Sydney metropolitan area.
NSW New South Wales other than Sydney
Melbourne The Melbourne metropolitan area.
VIC Victoria other than Melbourne.
BrisbaneGC The Brisbane and Gold Coast area.
QLD Queensland other than Brisbane and the Gold Coast.
Capitals The other five capital cities: Adelaide, Hobart, Perth, Darwin and Canberra.
Other All other areas of Australia.

Fig::Australia Vistor Nights Heirarchy
Glimpse of the Data at Hand
Sydney NSW Melbourne VIC BrisbaneGC QLD Capitals Other
1998 7320 21782 4865 14054 9055 8016 9178 10232
1998.25 6117 16881 4100 8237 5616 8461 6362 9540
1998.5 6282 13495 4418 6731 8298 13175 7965 12385
1998.75 6368 15963 5157 7675 6674 9092 6864 13098
1999 6602 22718 5550 13581 9168 10224 8908 10140
1999.25 5651 14775 3902 7883 7351 9672 7690 9948

Intuition

I promise you guys will try to keep the focus more on the intuitive part relative to the theory bu more advanced/curious folks can refer here

Summary of the 3 methods broadly which can be used to forecast hierarchical and grouped time series.

Summary Forecast Methods

Forecast Method Description
bottom-up approach This approach involves first generating base independent forecasts for each series at the bottom level of the hierarchy and then aggregating these upwards.
top-down approach Top-down approaches involve first generating base forecasts for the “Total” series top of the hierarchy and then disaggregating downwards.
Middle-out approach A hybrid of the above two approaches.

Briefly wiill go try and explain the above methods in our context in laymanish terms.

  • bottom-up approach: Here this means we will forecast lower most level of the Heirarchy i.e cities and then aggreagte the results up the heirarchy.

  • top-down approach: Here this means that we will forecast at the highest level heirarchy i.e pan Australia level and then disaggregate the results down the heirarchy.This disaggregation can happen via
    • computing relative proportions historically and using that for future periods.
    • forecasting the proportions in isolation,normalising it and then using that to disaggregate.
  • Middle-out approach:It’s a hybrid approach.Basially in this:
    • A middle level is chosen let’s say the state level which has 4 series/nodes within it.
    • 4 time series models at this level will be made.
    • For the top mode level bottom up approcah would be used to aggregate the results.
    • For the bottom level middle level results would be disaggregated downwards.

Pretty simple right!.

Now implementing this is even simpler.I will demostrate a working example via one of the above techniques and rest you can carch up via the documentaion of hts package in R language.

Application

y <- hts(vn, nodes=list(4,c(2,2,2,2)))

The above command creates a heirarchical time series with 3 levels(top most level one does not have to specify) with 4 nodes in the middle and 8 nodes in bottom most level.(basically the argument ‘nodes’ does the trick for you here.)

allf <- forecast(y, h=8,method = 'tdfp',fmethod = 'ets')#here tdfp means top-down forecast #proportions
plot(allf)
Forecasts across all the Heirarchies

The above command will give you forecasts across all levels in the heirarchy using top-down forecast proportions approach discussed above.

Just to reiterate:

  • Level 0–pan Australia level
  • Level 1–State level
  • Level 2–City Level

One can go futher deep in the documentation here and try out different techniques with different parameter combinations.